{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:30:31Z","timestamp":1777703431239,"version":"3.51.4"},"reference-count":25,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2019,6,18]],"date-time":"2019-06-18T00:00:00Z","timestamp":1560816000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"published-print":{"date-parts":[[2019,7,9]]},"abstract":"<jats:p>\n                    In this paper, we introduce the notions of node, nodal filter and seminode in equality algebras and study some properties of them. First, we study the relation between nodes and other specific elements. Furthermore, by defining some operations on\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                        <mml:mi mathvariant=\"script\">F<\/mml:mi>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mi mathvariant=\"script\">E<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    , which is the set of all nodal filters in an equality algebra\n                    <jats:italic>E<\/jats:italic>\n                    , we prove that\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                        <mml:mi mathvariant=\"script\">F<\/mml:mi>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mi>E<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                        <mml:mo>,<\/mml:mo>\n                        <mml:mo>\u2227<\/mml:mo>\n                        <mml:mo>,<\/mml:mo>\n                        <mml:mo>\u2194<\/mml:mo>\n                        <mml:mo>,<\/mml:mo>\n                        <mml:mi>E<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    is an equality algebra. In fact, we show that\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                        <mml:mi mathvariant=\"script\">F<\/mml:mi>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mi mathvariant=\"script\">E<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    is a Hertz algebra, BCK-algebra, Hilbert algebra, Kleene algebra and semi-De Morgan algebra. Then we investigate the relation among nodal filters and (positive) implicative, fantastic, prime, and boolean filters in any equality algebras. Finally, we study the relation between nodes and seminodes. And we prove the set of all seminodes\n                    <jats:italic>SN<\/jats:italic>\n                    (\n                    <jats:italic>E<\/jats:italic>\n                    ) is a lattice, Heyting algebra and Hertz algebra under the conditions.\n                  <\/jats:p>","DOI":"10.3233\/jifs-182860","type":"journal-article","created":{"date-parts":[[2019,6,21]],"date-time":"2019-06-21T11:50:42Z","timestamp":1561117842000},"page":"1457-1466","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":0,"title":["Nodal filters and seminodes in equality algebras"],"prefix":"10.1177","volume":"37","author":[{"given":"Xin","family":"Xun","sequence":"first","affiliation":[{"name":"School of Mathematic, Northwest University, Xi\u2019an, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiao Long","family":"Xin","sequence":"additional","affiliation":[{"name":"School of Mathematic, Northwest 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